Bayesian Hierarchical Models for Data Extrapolation and Analysis in Rare and Pediatric Disease Clinical Trials

Transcription

1 Bayesian Hierarchical Moels for Data Extrapolation an Analysis in Rare an Peiatric Disease Clinical Trials A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Cynthia Basu IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Avise by Braley P. Carlin, Ph.D. August, 2017

2 c Cynthia Basu 2017 ALL RIGHTS RESERVED

3 Acknowlegements I must begin by expressing my sincere gratitue to my acaemic a, Prof. Braley P. Carlin, to whom I am forever in ebt for his relentless patience, guiance, support, knowlege, an motivation. It was truly an honor to have gotten to know him not only as an astouning researcher, but as a wonerful human being an the most amazing avisor. I woul also like to thank my committee members Prof. Lin Zhang, Prof. Jim Hoges an Prof. Jim Cloy not only for their support but also their invaluable inputs that wiene my perspective on this work an helpe me better unerstan my own topic. I am also grateful to all my collaborators across the years, without whom none of this work woul have been possible. I am thankful to my pharmacology collaborators who helpe me comprehen an appreciate their subject. Their expertise an support is what enable me to pursue problems in the area of pharmacokinetics/pharmacoynamics. I am especially grateful to my statistical collaborators at Amgen an the DIA Bayesian Statistics Working Group for all their har work that went into eveloping these concepts. I am also thankful to my supervisors at the FDA for mentoring me for three months over the summer of 2015 an helping me better appreciate the practice of sensible regulation of clinical trials. Finally, my heartfelt gratitue to my friens who always stoo by me an mae this journey more joyful. My fellow stuents in the Division of Biostatistics an at the university in general provie feeback, cooperation, an most importantly their frienship. Most of all I am thankful to my parents for believing in me an instilling in me their faith in eucation an knowlege. i

4 Deication To my mother an father, Lucy an Sanjay Basu. ii

5 Abstract A rare isease is efine by the Rare Diseases Act of 2002 as a isease that currently affects fewer than 200,000 patients in the USA. A peiatric population is one where the subjects are of age 18 or less. These two crucial yet unerserve types of populations come with their own limitations in clinical trials. The paucity of potential trial enrollees an sensitivity of these patients, combine with a lack of sufficient natural history an experience, presents several economical, logistical an ethical challenges when esigning clinical trials. An increasingly well accepte approach to aress these challenges has been ata extrapolation; that is, the leveraging of available ata from aults or oler age groups to raw conclusions for the peiatric population. Bayesian hierarchical moeling facilitates the combining (or borrowing ) of information across isparate sources, such as ault an peiatric ata. In this thesis we begin by eveloping, illustrating, an proviing suggestions for Bayesian statistical methos that coul be use to esign improve clinical trials for peiatric an rare isease populations that efficiently use all available information. A variety of relevant Bayesian approaches are escribe, several of which are illustrate through two case stuies: extrapolating ault binary efficacy ata to expan the labeling for the rug Remicae R to inclue peiatric ulcerative colitis, an a simulate continuous longituinal ataset patterne after an evaluation of the rug cinacalcet in treating peiatric seconary hyperparathyroiism (HPT). The thesis then turns to methos useful in the stuy of X-linke arenoleukoystrophy (X- ALD), a rare neuroegenerative isease for which Lorenzos Oil (LO) is one of the few available treatments. We offer a hierarchical Bayesian statistical approach to unerstaning the pharmacokinetics (PK) an pharmacoynamics (PD) of LO, linking its ose to the plasma erucic aci concentrations by PK moeling, an then linking this concentration to a biomarker (C26, a very long-chain fatty aci) by PD moeling. Next, we esign a Bayesian Phase IIa stuy to estimate precisely what improvements in the biomarker can arise from various LO oses while simultaneously moeling a binary toxicity enpoint. Our Bayesian aaptive algorithm emerges as reasonably robust an efficient while still retaining goo classical (frequentist) operating characteristics. Future work in this area looks towar using the results of this trial to esign a Phase III stuy linking LO ose to actual improvements in health status, as measure by the appearance of brain lesions observe via magnetic resonance imaging. Finally, the thesis shows how to utilize the rich PK/PD ata to inform the borrowing of information from aults uring peiatric rug evelopment. Here we illustrate our approaches in the context of evaluating safety an efficacy of cinacalcet for treating HPT in peiatric an iii

6 ault patients. We use population PK/PD moeling of the cinacalcet ata to quantitatively assess the similarity between aults an chilren, an in turn use this information in various hierarchical Bayesian rules for borrowing from aults, statistical properties of which can then be evaluate. In particular, we simulate the bias an mean square error performance of our approaches in settings where borrowing is an is not warrante to inform guielines for the future use of our methos. iv

7 Contents Acknowlegements Deication Abstract List of Tables List of Figures i ii iii viii x 1 Introuction Data Extrapolation in Rare an Peiatric Diseases Datasets Use in the Thesis Remicae for Peiatric Ulcerative Colitis Lorenzo s Oil for X-ALD Cinacalcet for Peiatric Seconary HPT Hierarchical Bayesian Moels for Clinical Data Extrapolation Backgroun Classical Statistical Approaches to Data Extrapolation Current Bayesian Approaches Binary Response Moel Two-step Approach Combine Approaches Continuous Longituinal Response Moels Two-Step Approach Combine Approaches Outlook v

8 3 Bayesian Moels for PK/PD an Phase IIa Stuies in Rare Diseases Motivating Example in Rare Disease Hierarchical Bayesian PK/PD Moelling The Pharmacokinetic Moel The Pharmacoynamic Moel Results for the LO PK/PD Data Phase IIa Design Approximate Linear PD Moel Emax PD Moel Phase IIa Emax Simulation Results Discussion PK/PD Data Extrapolation Moels for Improve Peiatric Efficacy an Toxicity Estimation Introuction an Motivating Dataset PK/PD Moeling Hierarchical Moelling of Clinical Efficacy Data Joint Clinical Efficacy an Toxicity Moel Simulation Stuy Piecewise Linear Moel for Efficacy Efficacy-only Moel Joint Efficacy an Toxicity Longituinal Moel: ipth an cca Discussion Conclusions an Future Work Summary Discussion Zellner s g-prior for Rare an Peiatric Disease Review of Zellner s g-prior Moifie Zellner s g-prior Simulate Datasets Choice of g Conclusion References 62 vi

9 Appenix A. 68 A.1 BUGS coe for Binary response moel using Power Prior for Section A.2 BUGS coe for PK/PD Analysis for Section A.3 BUGS Coe for Joint Clinical Efficacy an Toxicity Moel for Section vii

10 List of Tables 2.1 Stuy-level enpoint ata, Remicae UC stuies in aults (ACT 1 an ACT 2) an peiatrics (T72) Results for the various Bayesian moels fit to the stuy-level Remicae ata Posterior estimates for the moel coefficients using a two-step approach for various values of α Posterior estimates for the moel coefficients using the commensurate prior for various values of p Posterior estimates for the moel coefficients using the power prior for various values of λ Table epicting parameter estimates obtaine from the moel escribe Estimate power (stanar error) of the Phase IIa esign (column 4) an proportion of patients being assigne to each ose (columns 5-9) using the Emax moel for various choices of θ 3, β t, an n at µ t = 4 an θ 4i = for all i Posterior estimates for the moel coefficients using a power prior for varying values of λ = Kp. The three rows of the table correspon to K = 1, 0.1, an Posterior estimates for the joint moel coefficients using a power prior for varying values of λ Results from the simulation stuy assuming normal ranom effects Results from the simulation stuy assuming t 3 ranom effects Posterior Mean Estimates of Slopes of Percent Change in Intact Hormone Over Time in Piecewise Linear Regression Moel (Longituinal Moel) Using Commensurate Priors to Extrapolate Data from the Ault to the Peiatric Dataset Posterior Probability of True Mean Difference in Percent Change in Parathyroi Hormone for Cinacalcet Compare With Placebo at Monthly Time Points (Longituinal Moel) When Using Commensurate Priors to Extrapolate Information From Ault to Peiatric Data; viii

11 4.7 Posterior preictive probability of 30% reuction from baseline in ipth an posterior preictive probability of 8.4% in cca Posterior mean estimates of slopes of percent change in ipth over time an slopes for cca levels in the efficacy-toxicity longituinal moel The mean, stanar eviations an confience intervals for the moel parameters an g in ientical ault an peiatric atasets The mean, stanar eviations an confience intervals for the moel parameters an g in not similar ault an peiatric atasets ix

12 List of Figures 3.1 Left panel: plot of the PK ata (Y ij vs. estimate cssav ij ) along with the fitte regression line, which has slope 1.059; right panel: plot of the PD ata (Z ij vs. estimate cssex ij ) along with the fitte Emax curve Dosing algorithm for Phase IIa trial esign, patient i, bi-week j Plot epicting the power curve for the Emax moel at each value of n with increasing θ 3 for θ 4i = an µ t = 4 (low overall toxicity) an β t = 0.15 (low rug toxicity) The four compartment structure assume by the cinacalcet PK/PD moel Plot of the target function f(p) for various values of α. The right vertical axis correspons to Ψ = EHSS while the left vertical axis correspons to Ψ = EHSS/n c using the sample sizes from our cinacalcet ata Caterpillar plots of the posterior probability of toxicity (cca uner 8.4 an uner 7.5) an efficacy (more than 30% an 50% reuction in ipth from baseline)at week 24 of each peiatric patient in the rug group when using power λ = The piecewise linear moel s regression line for the placebo an cincalcet group, for an Iniviual with intercept η 0,i equal to zero Plots Showing the Prior an Posterior Densities of the Spike an Slab Prior an The Corresponing p In Them x

13 Chapter 1 Introuction 1.1 Data Extrapolation in Rare an Peiatric Diseases A rare isease is efine by the Orphan Drug Act of 1983 as a isease that currently affects fewer than 200,000 patients in the USA. Disease rarity not only reuces the possible caniates an cases available for stuy an research, but also limits patient natural history an experience available. A peiatric isease is one that affects iniviuals age 18 or less as efine by section 529(a)(3) to the Feeral Foo, Drug, an Cosmetic Act (the FD&C Act). Stuies involving rugs targete at peiatric or rare iseases require special attention, as they present several economical, logistical an ethical challenges 26,15,7. In the Unite States, legislative actions have been taken to promote research in these fiels, such as the Best Pharmaceuticals for Chilren Act 6, Peiatric Research Equity Act 44, an the Orphan Drug Act 43. As the populations involve are often more sensitive as well as fewer in number, we wish to minimize the risk involve while maximizing the little information we have. It woul be very helpful if we coul borrow information from all available ata sources, both to help substantiate our results from the current stuy an to attempt to reuce the neee sample size. Borrowing historical ata information has been well explore in regulatory science for meical evices, with guiance ocuments available from FDA 11,12. However, borrowing information in the case of rugs is a not as straightforwar, as the pharmacokinetics (PK) an the pharmacoynamics (PD) of the rug may be very ifferent in, say, aults an chilren, an hence the response from chilren may be ifferent from that of the aults 51. Also, rugs are often approve for an ault population an prescribe off-label to chilren with only minimal unerstaning of the vagaries of peiatric response. Later, the sponsors often wish to conuct new stuies to test the rug specifically in the peiatric population. In such stuies it is unethical to subject more chilren to experimentation than absolutely require, motivating use of the 1

14 2 ault ata from other clinical trials or any other historical ataset to strengthen the analysis an reuce the peiatric sample size. Many extrapolation techniques 10 have been use to streamline peiatric rug evelopment an help spee approvals of rugs for peiatric labeling. Bayesian statistical methos easily facilitate aaptive ose-fining an other early-phase stuies, permit formal borrowing of strength from other information sources incluing expert opinion an previous ata, an yiel probabilistic inference regaring moel quantities of interest. In this thesis we will look at Bayesian methos in several phases of clinical trials specifically relate to rare an peiatric iseases, with emphasis on ata extrapolation techniques to facilitate more efficient an ethical clinical trials. We begin Chapter 2 with a look at hierarchical Bayesian moels to evaluate an analyze Phase II-III rare an peiatric ata. This is a particularly crucial area for borrowing of information from historical or ault atasets, since there is a nee for Type I error control in these later, often regulate phases. We will look at various methos to borrow information, incluing power priors an commensurate priors. We will test methos using binary ata from clinical trials pertaining to the rug Remicae (infliximab), which was uner review by FDA as a treatment for peiatric unlcerative colitis. Next we formulate piecewise linear moels for continuous longituinal ata an show how ata extrapolation can be carrie out with such moels using a Bayes-Markov Chain Monte Carlo (MCMC) approach. We consier power an commensurate prior methos, as well as more stanar methos as summarize by Thompson an Pennello 53. We illustrate the methos to leverage ata using a simulate ataset patterne after a real ataset we escribe below. Next, in Chapter 3 we expan our attention to the pharmacokinetics an pharmacoynamics (PK/PD) stage, an expan our hierarchical Bayesian moeling to rug PK/PD. Here we are motivate by a ata set from a stuy on the effect of Lorenzo s Oil (LO) for patients with X- linke arenoleukoystrophy (X-ALD), a rare isease of primarily young boys. Using information from our PK/PD moelling, we esign a Phase IIa clinical trial for LO efficacy an toxicity that fine-tunes the ose aministration per patient s personal response. In our esign, we compare the use of stanar Emax an approximating linear PD moels, the latter of which can offer statistical computational benefits, especially when simulating trial operating characteristics. Chapter 4 then introuces a novel metho to utilize PK/PD ata to inform the amount of borrowing of information from previous ault ata uring the later clinical phases of peiatric rug evelopment. The basic iea here is that if the PK/PD properties of the rug are similar in aults an chilren, this helps justify borrowing more liberally from the ault ata in later phases of the peiatric evelopment program. Here again we tackle the case of longituinal continuous ata, an evelop moels for both efficacy alone, an efficacy an safety jointly. We

15 3 then use these moels to analyze a ataset on peiatric seconary hyperparathyroiism (HPT), where our power prior is calibrate using PK/PD ata from relate ault an chil stuies. We then perform a simulation stuy to help she light on the proper choice of calibration of our power prior. We close this chapter with preliminary results for a piecewise linear version of our longituinal moel, an some iscussion of alternate methos for measuring ault-chil similarity in the PK/PD ata. Finally, Chapter 5 conclues an summarizes the work. In particular, Section 5.1 focuses on irect extensions of the methos in Chapters 2-4, while Section 5.2 proposes a moifie version of Zellner s g prior an escribes its use in ata borrowing from historical an ault atasets. We also analyse simulate Gaussian ata base on the Remicae example using hierarchical Bayesian moeling, an again incorporating tools from borrowing methos such as power priors an commensurate priors. We also propose to evelop better methos for selecting g in this setting. 1.2 Datasets Use in the Thesis Remicae for Peiatric Ulcerative Colitis In Section 2.2 we consier a beta-binomial moel for binary response ata illustrate with a recent ataset on the rug Remicae R (infliximab), a rug approve by the FDA to treat Crohn s isease, ulcerative colitis (UC), rheumatoi arthritis, an other conitions. We have ata from two ault UC clinical trials, calle ACT 1 an ACT 2, as well as one peiatric UC clinical trial calle T72. Both ACT 1 an ACT 2 ha patients who were aministere infliximab at a ose 5mg/kg. The peiatric trial T72 ha only 60 peiatric patients who were aministere infliximab at 5mg/kg. This uneven sample size is often prevalent in such clinical trials, where the rug has been teste in aults previously an shoul be treate with caution. Our enpoint of interest here is base on the observe Mayo score which is a commonly use activity inex in placebo-controlle clinical trials for UC. The score is further elaborate in Chapter 2. The binary nature of the ataset as well as the uneven sample size across the trials present an interesting case stuy for the use of Bayesian ata extrapolation methos in clinical trials. The ataset is further elaborate in the chapter along with its analysis Lorenzo s Oil for X-ALD In Chapter 3, we analyze a PK/PD ata from young boys suffering from X-ALD. We are intereste in the effect of Lorenzo s Oil (LO) aministration on the change in C26:0 plasma concentration in these boys, while ajusting for various covariates specific to iniviuals an

16 4 observations, such as the ose of LO aministere, the weight of the patient, etc. LO is a 4:1 mixture of oleic an erucic aci, an aministration of LO is known to increase levels of erucic aci in the plasma. The analyze ataset contains ata from 116 subjects screene for this stuy at the John Hopkins Research Hospital from 2000 to 2014 uner an expane access trial (ClinicalTrials.gov, NCT ). Diagnosis of X-ALD was confirme by plasma SVLFCA assay. These asymptomatic X-ALD chilren were followe until they evelope any brain MRI abnormality. The patients returne to the clinic every few weeks for their bloo samples to be collecte, which resulte in one observation per visit. The lack of etaile observation per ose cycle of the patients presents a limitation to the usual PK/PD analysis. For the PK/PD analysis we are intereste in the patients erucic aci concentration, C26:0 level obtaine from the bloo samples an also each patient s weight in kg. Since the patients isplay elevate levels of C26:0, the average improvement in C26 level for each patient is the primary quantity of interest in our PD analysis as well as the Phase IIa trial. We look at further etails an analysis in Chapter Cinacalcet for Peiatric Seconary HPT We have two sets of ata from clinical trials for cinacalcet, namely a ataset for a PK/PD analysis, an a Phase III clinical trial ataset. Data for PK/PD Analysis The ataset for PK/PD analysis consists of ata from 7 clinical stuies, 3 of which are ault stuies (Amgen stuies , an ), an the remaining 4 peiatric stuies (Amgen stuies , , an ). The ault stuies inclue a Phase III ranomize controlle stuy with 403 patients ( ), a Phase II stuy with 60 patients (980126), an a Phase I stuy with 22 patients ( ). The peiatric stuies inclue 2 stuies with oler peiatric patients age 6-18 years ( , a Phase 3 ranomize controlle stuy with 43 patients, an , a Phase I stuy with 12 patients), an 2 stuies with younger peiatric patients age 28 ays to less than 6 years ( , a Phase 2 stuy with 11 subjects, an , a Phase 1 stuy with 12 patients). Due to the sequential nature of the PK/PD analysis explaine later in the section, we have the posterior estimates of the PK analysis necessary for the PD analysis. We also have extensive patient specific observations necessary for the PD analysis, incluing the ipth an cca observations which constitute the crucial feeback loop in the moel.

17 Data for Phase III Efficacy an Toxicity Analysis 5 The clinical efficacy an toxicity response ata are further analyze in Section 4.3. These ata came from three ault Phase III clinical stuies ( , an ) an one peiatric Phase III stuy ( ). The three ault stuies have similar stuy esigns an were ranomize, placebo-controlle clinical trials with subjects age at least 18 years with seconary HPT receiving ialysis. Total sample size in these ault stuies is N=1136. The peiatric ata were collecte from the 30-week, ouble-blin, placebo-controlle phase of peiatric stuy The stuy enrolle N=43 subjects age 6 to 18 years with seconary HPT receiving ialysis. During the course of the stuies, a cinacalcet ose was given aily, an labs incluing ipth an cca were measure every two weeks. The percent change in the level of ipth inicates the efficacy of the rug, whereas the rop in the level cca below a threshol value is an inicator of toxicity in the patient.

18 Chapter 2 Hierarchical Bayesian Moels for Clinical Data Extrapolation 2.1 Backgroun As mentione in Chapter 1, statistical borrowing of strength for auxiliary ata can offer a way forwar in rare an peiatric rug an evice approvals. In this chapter, we lay out such moels for both continuous an binary ata, an illustrate their application to two rare an peiatric isease settings Classical Statistical Approaches to Data Extrapolation The traitional approach to analyzing a trial for peiatric iseases, where the rug has alreay been approve for aults, is to carry out a stuy on chilren an analyze it without any information borrowe from ault ata. Depening on the type of ataset at han, as well as the backgroun clinical information, a wie variety of moels can be fit to explain the ose-response relationship. These coul be simple linear moels, ranom or mixe effects survival moels, piecewise linear regression moels, or logit moels for binary response atasets. Classical approaches inclue fitting simple frequentist ranom or mixe effects moels to just the peiatric trial ata, which is now routinely one using software packages such as R an SAS. An example for such a moel is given in (2.1), where we regress the mean response µ ij from the i th chil s j th observation on the time from baseline, enote here by T ime ij, the ose of the rug, enote by Dose ij, an introuce a subject-specific ranom intercept γ i : µ ij = γ i + β 1 T ime ij + β 2 Dose ij. (2.1) 6

19 For a continuous observation Y ij, a sensible likelihoo is 7 Y ij in N(µ ij, σ 2 Y ), i = 1,..., n c, j = 1,...m i, (2.2) for some σ 2 Y > 0 where n C is the number of chilren. However, this approach ignores the information in the ault ataset. An alternate but still naive approach woul be to pool all ault an chil observations into one big ataset an then fit a moel. This uncritical pooling of the peiatric an ault atasets may be inappropriate in many settings. A slight improvement to this proceure is to fit the regression once to the ault ata an again to the peiatric ata, an then take an appropriate weighte average of the peiatric an ault results. These weights coul be base on the sample sizes of the two atasets, or perhaps expert opinion about the rug an the isease. However, the weight selection is crucial but ifficult to justify in a traitional statistical framework. A key aspect of such problems is the basis upon which extrapolation is eeme appropriate. Several pieces in the literature have iscusse this issue 10, where a systematic review of approaches for matching ault systemic exposures is often use as the basis for ose selection in peiatric trials submitte to the FDA. The literature refers to two categories of extrapolation, full an partial extrapolation. Full extrapolation is when ault ata are use irectly to establish peiatric safety or efficacy. These extrapolations rely on ata supporting the assumptions that there are similar isease progressions, responses to intervention, an exposure-response relationships in the ault an peiatric populations. Peiatric evelopment supporte by peiatric pharmacokinetic an safety ata or peiatric safety ata alone can be consiere aequate. By contrast, partial extrapolation is when ault ata are statistically combine with peiatric ata to make such eterminations. Partial extrapolation of efficacy is use when there is uncertainty about at least one of the assumptions unerlying complete extrapolation, as mentione above. In such cases, peiatric evelopment coul be base on a PK-PD stuy to confirm response in the peiatric population, followe by a single, aequate, well-controlle peiatric trial to confirm the efficacy seen in aults Current Bayesian Approaches The approaches mentione so far work well when we have a fairly rich peiatric ataset an also unerstan the isease an rug mechanisms reasonably well. However when we are working with peiatric or rare iseases, we wish to use every bit of information available to us. In such cases, Bayesian methos can help strengthen our analysis through their ability to combine multiple information sources. Current Bayesian approaches inclue fitting hierarchical moels 5,8,30 to both the ault an peiatric ata together, where the parameters connecting the two atasets

20 are relate at some eeper level of the hierarchy. As an illustration, let us assume we have three atasets: two ault atasets D 1 an D 2, an a peiatric ataset D 0. Let the numbers of patients in each of these atasets be n 1, n 2, an n 0, respectively, an let i = 1,...n 1,...(n 1 + n 2 ),..., n enote the patient inex, where n = n 1 + n 2 + n 0 enotes the total number of patients. Then assuming a simple linear fixe effects moel like the one in (2.1), having likelihoo (2.2), we can construct the following hierarchical moel: µ ij = γ a,i + β a,1 T ime ij + β a,2 Dose ij for i = 1,..., (n 1 + n 2 ) µ ij = γ c,i + β c,1 T ime ij + β c,2 Dose ij for i = (n 1 + n 2 ) + 1,..., n, γ a,i N(γ 0a, σ 2 a), γ c,i N(γ 0c, σ 2 c ), γ 0a, γ 0c in N(γ 0, σ 2 γ) an β a,k, β c,k in N(β k, σ 2 β k ) for k = 1, 2. (2.3) We may procee to assign hyperpriors to γ 0, β 1, β 2, an the variance parameters base on prior information from other ata, such as crue auxiliary estimates of these numbers if possible, or using vague priors that let the ata irect the values of these parameters. Such hierarchical moels are wiely use an often yiel sensible results. Such moels can however be constructe to incorporate much more prior information than shown above, leaing the hierarchy to become more complicate an possibly rely too much on the ault ata. Viele 58 an Hobbs et al. 18 also stress that such static borrowing (where the amount of ault ata incorporate is fixe ahea of time) can lea to biase estimates when the two ata sources o not agree. Also, when we rely on vague hyperpriors, the moel may be computationally improper an our MCMC algorithm may fail to converge, thus renering the Bayesian approach futile. Care shoul thus be taken to incorporate all prior information an expert opinion regaring the problem at han, as well as to be reasonably parsimonious with parameters. Note also that all borrowing is implicit through the exchangeability of the β a,k an β c,k. Thus this metho is simple, but oesn t allow us to explicitly control the amount of borrowing between the atasets. An alternative to fitting an exchangeable moel for the atasets together is to take a twostep approach, where we first fit a hierarchical moel to the ault ataset, an then use its posterior estimates as the priors in a secon statistical moel for the peiatric ataset. This metho is illustrate in a real-life setting in Subsection We might also ownweight the ault information use by introucing various scaling parameters in the priors for the peiatric moel. In the longituinal continuous ata setting, suppose we first fit the ault moel to our ault ata an obtain posterior means an variances for γ 0a, β a,1 an β a,2, which we enote informally by ˆγ 0a, ˆσ 2 (ˆγ 0a ), ˆβa,1, ˆσ 2 ( ˆβ a,1 ), ˆβa,2, an ˆσ 2 ( ˆβ a,2 ). 8 Then in Step 2, we assume γ 0c N(ˆγ 0a, α 0ˆσ 2 (ˆγ 0a )), β c,1 N( ˆβ a,1, α 1ˆσ 2 ( ˆβ a,1 )), an similarly for β c,2 using α 2 to scale

21 9 the prior variance. The three hyperprior variance scaling parameters (α 0, α 1 an α 2 ) help us control the amount of borrowing from the ault ata. In other wors, we center our priors for these parameters at the posterior estimates we have obtaine for the aults, but ownweight this information by choosing α s bigger than 1. When these values equal 1, we assume full borrowing from the ault information. A slightly more sophisticate an flexible metho to facilitate ata extrapolation while retaining control on the amount of borrowing is through the use of the power prior 25. This approach ownweights the supplemental (ault) likelihoo by raising its likelihoo in the posterior calculation to a power that is between 0 an 1. Assuming we wish to regress our response variable on some inepenent variables an the parameter of interest is θ, the power prior base on two ault atasets is [ 2 ] π(θ, λ D 1, D 2 ) L(θ D k ) λ k π(θ), (2.4) k=1 where λ = (λ 1, λ 2 ) an the initial prior π(θ) is often vague. Note that λ k controls how much information will be borrowe from auxiliary (ault) ataset k to supplement the (fully-utilize) chil ata; e.g., λ k = 1 means full borrowing from source k, while λ k = 0 implies no borrowing from this source. Such control is important in cases where there is heterogeneity between the supplemental an primary ata, or when equal weighting of primary an all supplemental ata sources is inappropriate. In fact, for fixe power priors, there is a one-to-one relationship between the power parameter an the effective sample size in the prior. The relationship is particularly straightforwar in the normal likelihoo setting; see for example Morita et al. 33,34 an Penello an Thompson 45. Finally, the commensurate prior approach 18,19,20 is an even more fully aaptive metho to account for the commensurability between the ault an peiatric atasets. It essentially specifies a hierarchical moel with posterior p(θ c, θ a, η D 0, D 1, D 2 ) L(θ c D 0 )L(θ a D 1, D 2 )π(θ c θ a, η)π(θ a )π(η). (2.5) Hence the so-calle commensurate prior for the chil parameter vector θ c, π(θ c θ a, η), is usually centere aroun the corresponing parameter for the ault ata (e.g., N(θ c θ a, η 1 )). The amount of borrowing can be moifie by tuning the precision η of the prior aroun θ a. A larger variance woul imply we have less faith in the similarity of the peiatric an ault ata, an therefore allow the peiatric estimate to be farther from that of the ault. Hobbs et al. 20 recommen a spike an slab hyperprior for η, which helps crystallize the choice between borrowing an not borrowing. Both this metho an the power prior metho are also emonstrate in our Subsection case stuy.

22 ACT 1 (ault) ACT 2 (ault) T72 (peiatric) Infliximab 5mg/kg Infliximab 5mg/kg Infliximab 5mg/kg Enpoint n = 121 n = 121 n = 60 Clinical response 84 (69.4%) 78 (64.5%) 44 (73.3%) Clinical remission 47 (38.8%) 41 (33.9%) 24 (40.0%) Mucosal healing 75 (62.0%) 73 (60.3%) 41 (68.3%) 10 Table 2.1: Stuy-level enpoint ata, Remicae UC stuies in aults (ACT 1 an ACT 2) an peiatrics (T72) All Bayesian moels escribe above can be fit using stanar Bayesian software such as OpenBUGS 31, Stan (mc-stan.org), Proc MCMC in SAS, or in R using packages available in CRAN (cran.r-project.org) or those that call BUGS or its variants from R, such as rjags. 2.2 Binary Response Moel As iscusse in Section 1.2.1, we look at a beta-binomial moel for binary response ata on the rug Remicae R (infliximab). Its maker sought a meeting with a gastrointestinal (GI) FDA avisory panel for the purpose of expaning the rug s labeling to inclue peiatric ulcerative colitis. As is common in such settings, extrapolation from ault ata was not permitte for osing or safety assessment in chilren, but the panel i allow the sponsors to argue for extrapolation of efficacy using two existing ault stuies. Ultimately the panel i ecie in favor of full extrapolation, but no quantitative moeling was involve in this ecision, only clinical jugment. The summary statistics of the ata seen by the avisory panel have appeare in the literature 23,24. Recapitulating from Section 1.2.1, our ata comprises of K = 2 UC trials in aults (calle ACT 1 an ACT 2) an one UC trial in peiatrics (calle T72). The efficacy enpoints in these trials are base on the Mayo score erive from the subscores of its 4 components: stool frequency, rectal bleeing, enoscopic finings, an physician s global assessment. The Mayo score has a minimum value of 0 an a maximum value of 12. The primary efficacy enpoint of clinical response at week 8 is efine as a ecrease in the Mayo score by at least 30% an 3 points, with a ecrease in the rectal bleeing subscore of at least 1 point or a rectal bleeing subscore of 0 or 1. When this efinition is met, the clinical response is 1; otherwise, it is 0. The seconary enpoints are presence or absence of clinical remission an mucosal healing at week 8. Table 2.1 gives the summary statistics of the enpoints. In the remainer of this section, we apply Bayesian binary methos to these ata, obtaining quantitative summaries that might have helpe the panel make a better-informe ecision.

23 Two Step Prior Commensurate Prior Power Prior r (ESS) E(θ 3 D, D 0 ) (CI) κ α (ESS) E(θ 3 D, D 0 ) (CI) α 0 (ESS) E(θ 3 D, D 0 ) (CI) 0.01 (62.4) (0.616,0.832) 1 (62.4) (0.622,0.837) 0 (62) (0.615, 0.835) 0.25 (121) (0.607,0.761) 10 (84) (0.620,0.828) 0.25 (123) (0.631, 0.790) 0.5 (181) (0.602,0.728) 50 (181) (0.617,0.793) 0.5 (183) (0.638, 0.770) 1 (302) (0.606,0.711) 100 (302) (0.622,0.778) 1 (304) (0.648, 0.752) Table 2.2: Results for the various Bayesian moels fit to the stuy-level Remicae ata Two-step Approach Let Y jk enote the binary outcome on the primary enpoint for patient j in stuy k, an Y k = J j=1 Y jk enote the summary statistic for the primary enpoint total in stuy k, k = 1, 2, 3. Thus, the ault ata is D 0 = (Y 1, Y 2 ) an the peiatric ata is D = Y 3. We assume that Y k Binomial(n k, θ 0 ), for k = 1, 2 an use a conjugate Beta(κ a µ a, κ a (1 µ a )) prior on θ 0. We complete the moel specification by assigning hyperpriors κ a Uniform(2, 122) an µ a Beta(1, 1), where the upper boun for κ a was chosen to be comparable to the sample size of either of our ault atasets. The posterior istribution for a given κ a an µ a is Beta(κ a µ a + 2 k=1 Y k, κ a (1 µ a ) + 2 k=1 (n k Y k )). Note that the support of (κ a, µ a ) is (2, 122) (0, 1). In step 1, given only the ault ata D 0, enote the posterior means of κ a an µ a by ˆκ a an ˆµ a, an use these in the prior for the peiatric ataset. Specifically, we assume θ 3 Beta(rˆκ a ˆµ a, rˆκ a (1 ˆµ a )), where we use r (0, 1) to scale own the ault ata effective sample size (ESS), ˆκ a, as the peiatric population (n 3 =60) is much smaller than the combine ault populations (n 1 + n 2 = 242). Note that r can be either assume known or have a Beta(1, 1) prior; in the latter case is etermine from the ata. Here, we assume that r is known. In step 2 of this two step approach, we upate the Binomial(n 3, θ 3 ) likelihoo for Y 3 to obtain a Beta(rˆκ α ˆµ α + Y 3, rˆκ a (1 ˆµ a ) + n 3 Y 3 ) posterior for θ 3. Note that the empirical peiatric success rate is slightly higher than that of the aults, an hence when we borrow more strength from the ault ata, the θ 3 estimates shoul ecrease, an the corresponing 95% creible intervals shoul get narrower. The results for various choices of r (corresponing to ESS values ranging from 62.4 up to the full combine sample size of 302) are given in the first two columns of Table 2.2. Notice that as we increase the r, the θ 3 point estimate ecreases (from own to 0.659) an the corresponing 95% creible interval withs also ecrease (from own to 0.105), both as expecte. In this moel, the choice of r is somewhat subjective, but values greater than 0.5 result in 60/181 = 33% or less of θ 3 posterior s strength coming from the peiatric ata, which may be insufficient for regulatory approval.

24 2.2.2 Combine Approaches 12 In this section we first fit a commensurate prior moel. In particular, we choose a Beta(1, 1) istribution as our initial prior on θ 0 an take θ 3 θ 0 Beta(κθ 0, κ(1 θ 0 )) as our commensurate prior, with κ Gamma(κ α, 1) an κ α assigne a fixe value. The commensurate prior approach then specifies a hierarchical moel with posterior p(θ 3, θ 0, η D, D 0 ) L(θ 3 D)L(θ 0 D 0 )π(θ 3 θ 0, η)π(θ 0 )π(η). (2.6) Thus the joint posterior in this case arises as π(θ 3, θ 0,κ D, D 0 ) θ Y 3 3 (1 θ 3) n 3 Y 3 θ Y 1+Y 2 0 (1 θ 0 ) n 1+n 2 Y 1 Y 2 θ κθ (1 θ 3 ) κ(1 θ 0) 1 κ κα 1 e κ. (2.7) While this oes not lea to a close form for the marginal posterior p(θ 3 D, D 0 ), sampling from the istribution is routine via the BUGS language. The results for varying values of κ α are given in Table 2.2; the ESS values shown for this metho are compute as functions of our posterior estimates. As in the case with the two-step approach, increases in κ α are again associate with clear ecreases in the θ 3 point estimates an interval withs, though the shrinkage back to the ault values is less ramatic here. Finally, we also apply the power prior metho to our ata. Here we may begin by assuming that θ 0 = θ 3 = θ, instea of putting a homogeneity constraint on the θ k s. We then obtain the posterior p(θ D, D 0 ) as proportional to L(θ D)L(θ D 0 )π(θ). The power prior approach 25 ownweights the supplemental (ault) likelihoo by raising it to a power α 0 that is between 0 an 1. The power prior then arises as [ 2 ] π(θ, α 0 D 0 ) f(y k θ) α 0,k π(θ)π(α 0 ). (2.8) k=1 Note that α 0,k controls how much information will be borrowe from the auxiliary (ault) ata to supplement the fully-utilize chil ata; e.g., α 0,k = 1 means full borrowing from source k, while α 0,k = 0 implies no borrowing. Such control is important in cases where there is heterogeneity between the supplemental an primary ata, or when equal weighting of primary an supplemental samples is inappropriate. In fact, if the power priors are fixe, there is a one-to-one relationship between the power parameter of the power prior an the variance of the prior; the relationship is particularly straightforwar in the normal likelihoo setting. We fix the powers α 0,k = α 0 for k = 1, 2, thus specifying a fixe an equal amount of borrowing from both ault stuies. Notationally, we have Y k Binomial(n k, θ 3 ) an θ 3 Beta(κµ, κ(1 µ)), where we choose minimally informative values κ = 2 an µ = 0.5. Then

25 θ 3 D, D 0 Beta( κ µ, κ(1 µ)), where κ = κ + n k=1 n kα 0 an µ = κ 1 (κµ + Y K+1 + K k=1 Y kα 0,k ). The results for four representative values of α 0 (an corresponing ESS values) are given in the last column of Table 2.2. The effect of increasing ESS is again apparent, with the now-familiar trens in the posterior means an interval withs being somewhat intermeiary to those arising from the previous two methos Continuous Longituinal Response Moels Throughout this section our interest is riven by the cinacalcet ata escribe in Section Mimicking this ataset, we simulate atasets from both ault an peiatric clinical stuies of Cinacalcet in the context of a linear mixe effects Bayesian hierarchical moel to stuy the rug s effect on ipth. Cinacalcet has been shown to lower the parathyroi hormone (PTH) release by the parathyroi glans, which in turn reuces the level of calcium an phosphorous release from the bones. The ipth (intact PTH) test level is of key interest, an is routinely monitore for people with chronic kiney isease; lowering it by some clinically significant percentage is a goal of many efficacy trials in this area. The number of patients in the peiatric ataset is assume to be just n c = 40, whereas we suppose there are n a = 800 patients in the ault stuy, a level of imbalance not uncommon in practice. Let X i,j be the percent change in the ipth level of patient i (i = 1,..., n c,..., (n c + n a )) in the week of the patient s j th observation, j = 1,..., m i (where m i varies from 3 to 25). That is: X i,j = ipth i,j baseline ipth i baseline ipth i 100. (2.9) This percentage change will be our outcome variable in the linear moel. Let t i,j enote the week after baseline for the j th observation on the i th patient. Since here we assume we on t have precise osing information for every patient, our moel (2.1) for the chilren becomes Xi,j c Normal(µ c i,j, 1/τe c ), i = 1,..., n c where µ c i,j = µ c 1it c i,j + I(rugi c = 1)(µ c tc i,j). (2.10) Here µ c 1i are the subject-level ranom effects, assume to inepenently follow a N(ηc 0, τ η) c specification, an µ c is the fixe effect of the rug on each chil s slope. Note we o not inclue intercepts in moel (2.11) since X ij is efine to be 0 at baseline (t ij = 0). Similarly for the aults, we assume X a i,j Normal(µ a i,j, 1/τ a e ), i = n c + 1,..., n c + n a where µ a i,j = µ a 1it a i,j + I(rug a i = 1)(µ a ta i,j). (2.11)

26 Now the µ a 1i are subject-level ranom effects, assume to inepenently follow a N(ηa 0, τ a η ) specification, whereas µ a is the fixe effect of the rug on the slope of the fitte ipth percent change variable. Regaring hyperpriors, both η c 0 an ηa 0 are assigne flat hyperpriors, µc an µa are assigne vague normal priors, an we place vague G(0.1, 0.1) hyperpriors on τ c e an τ a e Two-Step Approach We first fit a two-step moel along the lines of those escribe in Section Specifically, in Step 1 we fit the ault moel to our ault ata an obtain posterior means an variances for η a 0, an µ a, which we enote by ˆηa 0, ˆσ2 (ˆη 0 a), ˆµa, an ˆσ2 (ˆµ a ). Then in Step 2, we use these posterior estimates to guie our peiatric analysis. Specifically, we assume η0 c N(ˆηa 0, α 0ˆσ 2 (ˆη 0 a )) an similarly µ c N(ˆµa, α ˆσ 2 (ˆµ a )). The next question is therefore what the values of α 0 an α shoul be. They can be assigne base on expert knowlege, such as how similar clinicians think the two populations are likely to be an thus how much borrowing can be justifie. As mentione above, such static borrowing is straightforwar but clearly somewhat subjective. Alternatively, we can assign hyperpriors to the α s, such as a vague gamma or spike an slab istribution. However in some cases this may not be a goo iea as it is often ifficult to specify this hyperprior, an no information in our ata exists to inform this ecision. A quantity sometimes use to guie this ecision is the effective historical sample size (EHSS) 19, which generalizes our notion of ESS in the beta-binomial case of Section 2.2. Various efinitions exist, but a straightforwar one 53 for a parameter of interest ξ is [ V ar(ξ X c ] ) EHSS(ξ) = n c V ar(ξ X c, X a ) 1, (2.12) which is the percent improvement in ξ s posterior precision (inverse variance) arising from using the chosen fraction λ of the ault ata, expresse on the same scale as the chil sample size. Note that other metrics (e.g., V ar 1 ) might have been chosen in (4.1), an that the answer will vary with the choice of ξ. In our case, we take the overall fitte slope in the rug group, η0 c + µc, as the primary parameter of interest ξ. Ieally the effective historical sample size shoul be no greater in magnitue than the peiatric sample size, since even though we are trying to borrow strength from the ault ata, our analysis shoul be primarily riven by the peiatric ata. However for our simulate ata we often see EHSS values approaching or even exceeing the actual ault sample size of 800, since efinition (4.1) is simple but may perform erratically in more complex hierarchical moels, especially when implemente via MCMC. Tables contain the results of the methos applie to our simulate ata. We look at the estimate posterior mean an stanar eviation (s), the 95% equal tail Bayesian creible

27 η0 c η0 c+µc BCI BCI BCI BCI α mean s lower upper EHSS mean s lower upper EHSS Table 2.3: Posterior estimates for the moel coefficients using a two-step approach for various values of α. interval (BCI), an the calculate EHSS corresponing to the placebo effect, η0 c, an the overall slope in the treatment group, η0 c +µc. The latter inicates whether the patients in the treatment group showe improvement over time. In Table 2.3 we fit our two-step moel using three ifferent values of α = α 0 = α to show the effect of borrowing. The posterior estimates from Step 1 were ˆη 0 a = an = Note that as α increases, the estimates for the chilren become more issimilar ˆµ a to those of the aults. Note also that for the vaguest prior (α=100), we obtain the chil ata-only results (EHSS=0), which also have the largest estimate s s. It appears for aroun α = 15, EHSS(η c 0 + µc ) is fairly close to the actual size of the peiatric ataset (n c = 40). It shoul also be note that base on the upper limit of the BCI for η0 c + µc, for example, the significance of our finings for the primary parameter change with α. In our case the change happens aroun α = 7; smaller values lea to statistically significant finings, whereas larger values o not Combine Approaches In this subsection we begin by fitting a commensurate prior moel. We assume our moel for the aults is as efine previously. We also moel the peiatric ata similar to before, but moify its prior to aaptively learn from the ault ata base on their estimate similarity. Following (2.6), we assign the prior for η0 c as N(ηa 0, 1/τ c), an the prior for µ c as N(µa, 1/τ c) to introuce the commensurability. We then procee to assign spike an slab hyperpriors on τ c an τ c as follows: { Normal(200, 0.01) with probability f; τ c Uniform(0.1, 5) with probability 1 f, { Normal(200, 0.01) with probability f; an τ c Uniform(0.1, 5) with probability 1 f,

28 η0 c η0 c+µc BCI BCI BCI BCI p mean s lower upper EHSS mean s lower upper EHSS Table 2.4: Posterior estimates for the moel coefficients using the commensurate prior for various values of p. η 0 η 0 +µ BCI BCI BCI BCI λ mean s lower upper EHSS mean s lower upper EHSS Table 2.5: Posterior estimates for the moel coefficients using the power prior for various values of λ. where we assume a single spike probability f Bernoulli(p) for both τ c an τ c. We can now vary the amount of borrowing by varying the value of p. An increase in p woul mean a higher chance of the precision taking a value close to 200, our spike, an hence more borrowing from the ault ata. By contrast, small values of p encourage small τ values in the slab, which iscourages ault borrowing. Relatively little information on τ c an τ c exist in the ata, so the spike an slab parameters must be tune carefully. Table 2.4 contains the results of this moel for varying values of p. As can be seen, the results show the expecte trens regaring borrowing between the atasets. again get values consistent with no borrowing (EHSS=0). For p = 0, we The EHSS(η c 0 + µc ) inicates a p of 0.65 elivers an EHSS approximately the size of the peiatric ataset. However the EHSS oes increase greatly for p > 0.9, becoming more than the available number of ault patients (n a = 800). As note in the case of the two-step approach, the significance of our finings changes with an increase in p, reemphasizing the caution with which the value of these parameters shoul be chosen. For η0 c + µc, the change seems to occur aroun p approximately 0.9; the BCI contains zero for smaller p, an oes not contain zero for very large p. Finally we fit a power prior moel. Here we may begin by assuming that η a 0 = ηc 0 = η 0 an µ a = µc = µ. We then obtain the posterior p(µ D a, D c ) as proportional to L(µ D c )L(µ D a ) λ π(µ ).